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1. B Complex Replacement: Justification?

Hi, Is there a proof that complex replacement is a valid way to solve a differential equation? I'm lacking some intuition on the idea that under any algebraic manipulations the real and imaginary parts of an expression don't influence each other. For example, if I'm given: $$p(D) x = cos(t)$$...
2. I Undamped Driven Oscillation

The equations I'm getting when I solve the differential equations seem to imply that the amplitude of oscillation does not vary in time. For example, if I have x'' + ω02x = cos(ωt) If we suppose that ω≠ω0, then the general solution should look something like: x(t) = c1cos(ω0t) + c2sin(ω0t)...
3. I Undamped Driven Oscillation

The equations I'm getting when I solve the differential equations seem to imply that the amplitude of oscillation does not vary in time. For example, if I have x'' + ω02x = cos(ωt) If we suppose that ω≠ω0, then the general solution should look something like: x(t) = c1cos(ω0t) + c2sin(ω0t)...
4. I Conceptual Question: Vector-Matrix Differential Equation

Hi I'm just having trouble wrapping my head around differential equations with matrices and vectors... For example: let y be a vector. let A(t) be an nxn matrix. I have the differential equation: dy/dt = A(t)y So I think I understand why the solution is y = ceA(t) But I'm having trouble...
5. Limit (sin(4x)/sin(6x)) as x->0

Hi all, I'm just beginning calculus and I'm having trouble figuring this one out. I need to solve this one without using l'hospital's rule. 1. Homework Statement Find Limit (sin(4x)/sin(6x)) as x->0 2. Homework Equations We know that limit (sin(x)/x) as x -> 0 equals...
6. Mirau White Light Interferometer. Light Source?

Hi everyone, I'm building a white light interferometer and need help choosing and finding a white light source. I'm using this mirau objective: http://www.edmundoptics.com/microscopy/infinity-corrected-objectives/nikon-interferometry-objectives/59312 Here is the setup: I know...
7. Do I understand the Fizeau interferometer? How to read interferogram?

Hi, I'm just wondering if someone could help me better understand how the Fizeau interferometer works... http://upload.wikimedia.org/wikipedia/commons/2/2f/Fizeau_interferometer_testing_optical_flat.svg From my understanding, the source shines light through a collimating lens and the light...
8. Polar Coordinates - Graphing the points of when theta<0

Polar Coordinates --- Graphing the points of when theta<0 Hi everyone, I'm working with an online graphing program desmos.com. It's great, and actually tons of fun. I'm currently working with polar coordinates but the only flaw of this grapher is that when working with polar coordinates...